Sydney Opera House-SOSO curves with a million fitted tiles! Photo by Arvin Wiyono on Unsplash

Polynomial Roots — Applying Curved Segment transplants from ‘SOSO’ Nodes

Using the ‘architecture’ to find roots from confected ‘SOSO’ nodes with transplanted segments having the same initial gradients as the recipient root segments.

Since I presented Cubic Polynomials — A Simpler Approach to find the last two roots of any polynomial given the other roots, my focus has been on improving node coordinates for application of Newton’s Approximation in finding first roots.

While exact lengthy numerical methods and calculators are of course available, I prefer to use ‘visual architecture’ as it aids learning, while also finding good approximate solutions. My objective in this post is twofold:

• to simplify and improve initial node coordinates by confecting twin Same Opposite Same Opposite, or ‘SOSO’ functions with intercepts, and;
• to replace Newton’s straight line hypotenuse approximations using Segment Transplants Incorporataing Node Gradients, or ‘STING’. It is a transplanted curved segment with the same node gradient as a root segment.

This post assumes knowledge of algebra and introductory calculus (differentiating polynomials) at the high school level.